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Properties

Label 2-960-5.3-c2-0-18
Degree $2$
Conductor $960$
Sign $-0.181 - 0.983i$
Analytic cond. $26.1581$
Root an. cond. $5.11449$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.22 + 1.22i)3-s + (4.99 + 0.248i)5-s + (−6.09 + 6.09i)7-s + 2.99i·9-s − 0.229·11-s + (−1.45 − 1.45i)13-s + (5.81 + 6.42i)15-s + (16.2 − 16.2i)17-s + 9.67i·19-s − 14.9·21-s + (25.8 + 25.8i)23-s + (24.8 + 2.48i)25-s + (−3.67 + 3.67i)27-s + 40.1i·29-s − 46.3·31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (0.998 + 0.0497i)5-s + (−0.870 + 0.870i)7-s + 0.333i·9-s − 0.0208·11-s + (−0.112 − 0.112i)13-s + (0.387 + 0.428i)15-s + (0.957 − 0.957i)17-s + 0.509i·19-s − 0.711·21-s + (1.12 + 1.12i)23-s + (0.995 + 0.0992i)25-s + (−0.136 + 0.136i)27-s + 1.38i·29-s − 1.49·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.181 - 0.983i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.181 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $-0.181 - 0.983i$
Analytic conductor: \(26.1581\)
Root analytic conductor: \(5.11449\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1),\ -0.181 - 0.983i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.167121405\)
\(L(\frac12)\) \(\approx\) \(2.167121405\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.22 - 1.22i)T \)
5 \( 1 + (-4.99 - 0.248i)T \)
good7 \( 1 + (6.09 - 6.09i)T - 49iT^{2} \)
11 \( 1 + 0.229T + 121T^{2} \)
13 \( 1 + (1.45 + 1.45i)T + 169iT^{2} \)
17 \( 1 + (-16.2 + 16.2i)T - 289iT^{2} \)
19 \( 1 - 9.67iT - 361T^{2} \)
23 \( 1 + (-25.8 - 25.8i)T + 529iT^{2} \)
29 \( 1 - 40.1iT - 841T^{2} \)
31 \( 1 + 46.3T + 961T^{2} \)
37 \( 1 + (45.9 - 45.9i)T - 1.36e3iT^{2} \)
41 \( 1 - 33.1T + 1.68e3T^{2} \)
43 \( 1 + (-13.1 - 13.1i)T + 1.84e3iT^{2} \)
47 \( 1 + (8.86 - 8.86i)T - 2.20e3iT^{2} \)
53 \( 1 + (57.1 + 57.1i)T + 2.80e3iT^{2} \)
59 \( 1 - 24.4iT - 3.48e3T^{2} \)
61 \( 1 - 23.9T + 3.72e3T^{2} \)
67 \( 1 + (44.2 - 44.2i)T - 4.48e3iT^{2} \)
71 \( 1 - 96.7T + 5.04e3T^{2} \)
73 \( 1 + (-6.78 - 6.78i)T + 5.32e3iT^{2} \)
79 \( 1 - 78.7iT - 6.24e3T^{2} \)
83 \( 1 + (60.8 + 60.8i)T + 6.88e3iT^{2} \)
89 \( 1 + 3.67iT - 7.92e3T^{2} \)
97 \( 1 + (-123. + 123. i)T - 9.40e3iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.800041474688666737003757285076, −9.349092168241208684207430388480, −8.742597666073560779513938656493, −7.49438899834148679271579091527, −6.63003768957991854202625394962, −5.50885109825638227432645892949, −5.15842158202968020963202204358, −3.39824154165581042130995365094, −2.85786818413728177624429129741, −1.53263798876440078747452989114, 0.64970327759561318857281858186, 1.95900482777732987287763286396, 3.08015275905861456441378944682, 4.07332334383448054999864084713, 5.39263309708830040526337759145, 6.30950464422943261866768651647, 6.98068165616020570975179814049, 7.83098754278719208829115037455, 8.975486226109911893973480187087, 9.497801656140930996038676751250

Graph of the $Z$-function along the critical line